Question

Suppose that $f^{\prime \prime }(x)$ is continuous for all $x$ and $f(0)=f^{\prime }(1)=1$ If $\underset{0}{\overset{1}{\int }}$ tf${}^n$ $(t)dt=0,$ then the value of f(1) is

• A. 2
• B. 3
• C. 4 $\frac{1}{2}$
• D. none of these

Answer 1

Answer: (A)

Since $\underset{0}{\overset{1}{\int }}t{f}^{\prime }\left(t\right)dt=0$
$\therefore {\left|t{f}^{\prime }\left(t\right)\right|}_0^1-\underset{0}{\overset{1}{\int }}1f^{\prime }\left(t\right)dt=0$
$\Rightarrow f^{\prime }\left(t\right)-\underset{0}{\overset{1}{\int }}{f}^{\prime }\left(t\right)dt=0$
$\Rightarrow f^{\prime }\left(1\right)-{\left|f\left(t\right)\right|}_0^1=0$
$\Rightarrow f^{\prime }\left(1\right)-\left[f\left(1\right)-f\left(0\right)\right]=0$
$\Rightarrow f^{\prime }\left(1\right)-f\left(1\right)+f\left(0\right)=0$
$\Rightarrow f\left(1\right)=f^{\prime }\left(1\right)+f\left(0\right)=1+1=2$